Support of a distribution ($\mathrm{supp}u$)

The support of a distribution is the complement of the largest open set on which its pairing vanishes for every supported test function. A partition of unity shows that the union of such open sets still has this vanishing property. For example, the Dirac delta distribution and all its derivatives have support $\{0\}$.

A distribution has compact support when it vanishes on every test function supported outside some compact set. Every compactly supported distribution has finite order.

A compactly supported distribution of order of a distribution at most $M$ has a smooth function as its Fourier transform, bounded by $C⟨\xi {⟩}^M$. It therefore belongs to every Sobolev space $H^s$ with $s<-M-n/2$.

For $u\in {\mathcal{E}}^{\prime }({\mathbb{R}}^n)$, its Fourier transform is the smooth functionIt has at most polynomial growth, and its extension to complex frequency is controlled more precisely by the Paley–Wiener–Schwartz theorem.

Every compactly supported distribution is a finite sum of distributional derivatives of bounded continuous functions. One proof convolves it with a sufficiently high-order Bessel potential and then applies a power of $1-\Delta$.

The Bessel potential of order $s$ is the Fourier multiplier $(1+|\xi {|}^2{)}^{-s/2}$. Sufficiently high order turns a compactly supported finite-order distribution into a bounded continuous function.